Lotka-Volterra Predator-Prey Models

1 Lotka-Volterra Predator-Prey Models by Danna Rammal Student Number: 0600936 Institute: Trent University Course: MATH- 4800H Honours Project Date: Monday, April 6th, 2020 2 Table of Contents: Chapter 1: 3 1.1 Abstract 3 1.2 History on the Lotka-Volterra Model 4 1.3 INTRODUCTION 5 Chapter 2 7 2.1 Relationship of variables in the Lotka-Volterra Model 7 2.2 Derivation of Formulas 8 2.3 Graphical representation of the Predator-Prey Model 1​4 2.4 Conclusion 2​5 References ​26 3 Chapter 1: 1.1 Abstract A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. It’s not only used in mathematics, but also in physics, computer science, economics, and so on. The Lotka-Volterra model is a widely used pair of first-order nonlinear differential equations used to interpret the dynamics of two species: a predator and a prey. The model is also known as the predator-prey equations as it mainly studies the behaviour and interactions of the two species in ecology. It uses various mathematical concepts such as linear algebra, calculus, as well as ordinary differential equations to implement the constraints into real life situations. 4 1.2 History on the Lotka-Volterra Model The Lotka-Volterra model was initially proposed by Alfred Lotka to analyse predator-prey interaction in his book on biomathematics. Alfred Lotka was an American mathematician who was interested in the behaviour of species preying on or being preyed by other species. In 1925 he began to study the interactions between species and realized that various species that would compete for a common source of food created a different category in his studies. Vito Volterra was an Italian mathematician who was widely known for his field of research and interest in mathematical biology. While Volterra was interested in understanding the competitive interactions between species, the two mathematicians proposed the term “ Lotka-Volterra equations”. Ecologically, the same title was used for the competition model, however it strictly applied only to predator-prey interactions. 5 1.3 INTRODUCTION The Lotka-Volterra model provides information about the competition between two​ species, living in the same ecosystem. To understand the lotka-volterra model, a few basic knowledge about ecology must be understood. If two species reside in the same habitat, then the same resources are shared between the two. Furthermore, the food habits as well as their food resources are common amongst them. Two terms ought to be defined in ecology: habitat, and ecological niche. A habitat is a place where an organism lives. It concludes with the abiotic factors in that habitat/place. While the ecological niche consists of the biotic factors, the food sources, as well as the habits of the organisms all together. Therefore, the niche is much more accurate and specific compared to the habitat, which only consists of biotic factors, and the biotic elements living there. Moving forward, the idea of a niche is referred to an organism’s functional role in the ecosystem that concluded the abiotic factor, the foods that organisms eat, the organism who preys on the other, as well as their behaviour. Ideally, with respect to a niche, no two species can coexist in the same niche without ending up in competition. Hence, if two 6 species share the same niche, then they ought to end up in competition. In ecology, that idea is known as the competitive exclusion principle or Gause’s Law. Gause’s Law states that if the niche is indifferent between two species, then the system will lead to a competition. Therefore, the result of the competition will be as follows: Competition → Win/Loss → Partitioning. Between these two species, if competition arises, one species will win and the other will lose. Furthermore, the one that succeeds will live in that habitat and the species that lose will be locally extinct. On the other hand, partitioning is a case such that both the species live and belong to the same niche together in the same habitat, but have limited resources. For example, consider two bird species that currently live in the same region of a plant and consume the same food, decide to live in two separate regions of the plant and eat different types of food. One bird feeds on the fruit, while the other on the seed. By the separation of food habits, the two can separate from their niche, to form a separate niche. This technique in specific will help prevent competition from ensuing between them. In this case, the first situation is of interest - when two species compete against one another. 7 Chapter 2 2.1 Relationship of variables in the Lotka-Volterra Model In order to predict the outcome between two species competing against each other, the Lotka-Volterra model will be used. Using this model in specific, the result of the condition will be determined based on the impact of one species on another. Starting with species X and Y, assume that both kinds live in the same region and are in competition due to having the same niche. Realistically, X will have an impact on Y, and Y will have an impact on X. The impact of X exerted on Y and vice versa is known as the competition coefficient, alpha ​α. ​Suppose that X and Y are taken to be species 1 and 2 respectively. Then, when X is present along with Y, species 1 is exerting 8 some competition on 2, known as alpha ​α​21​.​ Similarly, the amount of competition exerted by species 2 on species 1 will be ​α​12​. After having defined those terms, the determination of population growth can be analyzed for both kinds. If species 2 is being calculated, then the population of species 1 or species X must be taken into account as well. There is an inverse relationship between them because for instance, if the size of population for species 1 is larger, then the growth of species 2 will be less due to the competition. Thus, proceeding to the formula, the growth of the population can be calculated. In specific, the growth rate of a population can be calculated. 2.2 Derivation of Formulas The equation of the growth rate of population is as follows: dn dt Here, dn dt = rN ( 1.4.1) ​ ​is the total number of species increasing over time, N is the initial number of individual species present, and r is the intrinsic growth rate. The intrinsic growth rate is defined as the number of deaths subtracted from the number 9 of births per generation - in other words, the difference of reproduction rate and death rate. Graphing the equation would result in an exponential growth as such Fig 1. Image source: https://www.researchgate.net/publication/226819824_Ecotoxicological_Effects/figures?lo=1 The rapid growth is obtained from the growth rate of species, assuming that their resources are unlimited. An environment's ultimate resource that can support the length of a population is known as a carrying capacity. This factor refers to the total number of individuals a specific ecosystem can support for its growth, denoted as K. Once the carrying capacity has been taken into account for, a much 10 accurate growth rate will be obtained as a result. Therefore, introducing K to the equation gives dn dt = RN ( K−N K ) (1.4.2) The above equation will promote more precise data based on real situations of the wild as the carrying capacity has been introduced to the equation. In specific, a logistic growth, showing how populations actually grow, will be obtained as a result of graphing the formula above. Fig. 2 Image source: https://www.researchgate.net/publication/226819824_Ecotoxicological_Effects/figures?lo=1 11 Until this point, the general equations and graphs have been defined to calculate the population growth of one species. Below is the graph of the idealized exponential and logistic growth curves. Fig. 3 Image Source: PPK @ biology-forums.com In this graph above, the logistic and exponential growth initially correlate well, but eventually diverge from each other. The population growth exponentially increases continuously, with respect to an increase in the size of the population. Whereas the logistic growth curve increases in the population growth rather rapidly until reaching the carrying capacity. However, moving forward to a more complex 12 scenario, if one species is in competition with another species in the environment, then the two formulas (1.4.1) & ( 1.4.2) are not sufficient enough to determine the outcome. Therefore, a deeper understanding of the Lotka-Volterra model must be studied. If one species is in competition with another, then the population size of the other species as well as other factors must be taken into consideration. The Lotka-Volterra model is also called the predator-prey model because in a competition, it will be known that one of the species will win as the impact of each will be determined. A predator is an animal that naturally preys on others, whereas a prey is the animal being preyed upon. Typically, the species that will win in the competition will be the predator and the other is the prey, being less impactful. Once the total number of predators and prey in the population have been determined, then it will be clear as to what the outcome is going to be and what the growth rate will be of either the predator or prey in the environment. Therefore, in order to proceed with the derivation of the formula, the latter equation of 1.4.2 must be tweaked further. Assume that the total number of individuals present at the beginning of species 1 and species 2 is N​1​, N​2​ respectively. Then, the calculation for the growth of species 1 in the presence of species 2 will be dn dt = rN​1​ (1.4.3) 13 Thus, the equation to calculate the growth rate of species 1 is defined as S​1​: dn dt N2 = rN​1​( K−N 1−α12 ) K (1.4.4) Recall that K is the carrying capacity, α​21​ was subtracted from the total number of individuals for species 1 because the impact of one species on another is being taken into account. In this case, the growth rate of species 1 is being calculated, therefore, the impact is on species 1 by species 2 - hence, α​21​. ​In addition to that, N​2​ was multiplied by α​21​ ​because that is the total number of individuals present in species 2. Comparing equations 1.4.4 & 1.4.2, the two are almost indifferent, apart from the additional factor of ​α​12​N​2​. If the number of individuals in species 2 increased, then the impact from α​12​ will increase as well. Therefore, the total number of individuals present in species 2 must be taken into account for, which is why N​2​ was multiplied by. Similarly, a formula can be derived to obtain the growth rate of species 2 in the presence of species 1. S​2​: dn dt N1 = rN​2​( K−N 2−α21 ) K (1.4.5) 14 In both cases, the growth rate of species 1 and species 2 was able to be determined correctly, using the idea of the Lotka-Volterra model. Therefore it can be concluded that in reality, competition is always present, whether it is intraspecific of interspecific competition. To sum it up, it is always preferable to use the principle of the Lotka-Volterra model to accurately calculate the growth rate of one species in the presence of the other. 2.3 Graphical representation of the Predator-Prey Model Continuing on to the second part of the model, in this section, the behaviour of the species will be assumed and proved using mathematical graphs. As mentioned earlier, the outcome of the competition between two species can be predicted using the Lotka-Volterra model. In order to achieve that idea between both species and determine who is going to win and lose amongst them, graphs using the value of N​1​ and N​2​ must be constructed and analyzed. To perceive the predictions of the model, it is beneficial to analyze the graphs that demonstrate how the size of each population decreases or increases, starting with different combinations of species 15 abundances. Those include a high value in N​1​, low N​2​ or a low N​1​, low N​2​ and so on. On each graph that will be studied below, the x-axis and y-axis represent the abundance of species 1 and species 2, respectively. A zero isocline is a straight line politted on the graph for every species, that represents the abundances of the two species combined. For instance, the zero isocline for species 2 at an arbitrary given point constitutes of the two species together where species 2 in specific remains constant. Therefore the growth rate, dn dt = ​0 when the population of species 2 does not decrease or increase, and the value for N can be calculated. It is very important to calculate the carrying capacity of both species 1 and 2 in the presence of the opponent’s species. 16 Fig. 4 Image Source: ​https://www.liberaldictionary.com/isoclineisoclineisocline/ Observe that the graph above is divided in two proportions where the population size decreases on the right, and increases on the left of the isocline. The black arrows show trajectories of the change in population of species 1 and 2, whereas the multicolored dotted arrows represent the trajectories of individual populations in their respective species. In this case, the combination of the two abundances to the left is less than the carrying capacity, and larger than K on the right of the isocline. The carrying capacity is depicted in the y-axis as the maximum number of 17 individuals supported. Note that the isocline of the graph intersects at the x-axis, where no individuals of species 1 is present, resulting in N​2​ reaching its carrying capacity. While calculating the carrying capacity of K​2​, which is the carrying capacity of species 2, the presence of species 1 or the number of individuals present in species 1 must be taken into account, as well as ​α​12​. ​Recall that ​α​12​ ​is the impact of species 2 on 1, so to determine the carrying capacity of species 2, the number of species 1 and the impact of species 1 on species 2 must be calculated as well. If ​α​21​ ​is greater than ​α​12​, ​then this means that the impact of species 1 on 2 is far more greater than the impact of species 2 on 1. Thus in that case, the carrying capacity value will be less if the initial number of individuals present in species 1 is larger. Furthermore, the value of N​1​ will be high and the value for ​α​21​ is greater than ​α​12​, ​meaning that species 2 will lose and species 1 will win. This concludes the whole idea of the graphs using the Lotka-Volterra model. Using the same concept, if the value of ​α​12​ is greater than ​α​21​, and the initial number of individuals present in species 1 is less, then there is a far less impact coming from species 1 on species 2. Consequently, the impact provided by species 2 on 1 is high, therefore the value of the carrying capacity of K​2​ will reach a maximum and species 2 will win. This method of determining which of the two species will win or lose can be 18 used to understand and analyze the graphs. The population of the species is displayed for the cause when it is above or below one isocline at a time. In that event, three scenarios will be illustrated and studied for both species’s isoclines. Resultantly, the outcomes of the interspecific competition will possibly be predicted according to the relation of the isoclines with respect to the species. Fig. 5 Image Source adapted from Begon et al. (1996) & Gotelli (1998) 19 Similarly, for the zero isocline of N​1​, it reaches K​1​ on the x-axis in the absence of species 2. K​1​/​α​12 is ​ depicted on the y-axis where the value of the carrying capacity of species 1 is equivalent to N​2​, in the absence of N​1 Fig. 6 Image Source adapted from Begon et al. (1996) & Gotelli (1998) The first scenario is a graph of the calculation of parallel graphs for both species 1 and 2. The yellow colored line represents the isocline of species 1 and the growth pattern of species 1, meanwhile the pink dotted line represents the growth pattern of species 2. Focusing on the values of ​α​12​ ​and ​α​21​ ​in the parameters of K​1​/ ​α​12 20 and K​2​ ​/ α​21​, ​an inverse relationship between the carrying capacity and competition coefficients can be derived. Assuming that ​α​21​ ​is greater than ​α​12​, then the value of K​2​ ​/ α​21​ will decrease and that of K​1​/ ​α​12​ will increase. Therefore as a result, the carrying capacity for species 1, K​1​, will increase due to the low value of ​α​12​. Similarly, as ​α​21 ​increases, the carrying capacity of species 2 tends to decrease. The isocline for species 2 is below and to the left of species 1. Thus, any point in that lower left space will result in an increase in their respective isoclines for both species. Similarly, any point in the top right space will result in a decrease in both species as they are above their isoclines. Furthermore, If any point lies in between the two isoclines, then the isocline for species 2 will still remain on top, resulting in a decrease. Equivalently, species 1 will still remain below its isocline and increase for any point in between the two isoclines. Inspecting the graph of scenario 1, the carrying capacity of species 1 is the length along the x-axis until K​1​, and the carrying capacity for species 2 is the length along the y-axis until K​2​. It can clearly be seen that the length of K​1​ is longer than that of K​2​, meaning that with the same resources, the environment can support more of K​1​. In other words, the environment is favoring species 1, resulting in a win and an extinction of species 2. Stable equilibrium is reached on K​2​ on the y-axis, while species 1 continues to 21 increase up to the point where it reaches carrying capacity, K​1​. Using Gause’s Law in this case, species 2 will consistently be outcompeted by species 1. Fig. 7 Image Source: adapted from Begon et al. (1996) & Gotelli (1998) The same idea can be implemented to analyse the graph of scenario 2 as it is the opposite of the first where the isocline of species 1 is below species 2. In this case, α​12​ ​is greater than ​α​21​, ​resulting in a larger value of K​2​. Looking at the graph, the length of K​2​, which is the carrying capacity of species 2, is far greater than that of K​1​. From here, since K​2​ is greater than K​1​, species 2 will consistently outcompete 22 species 1. Note that more than one case used the same method to prove the prediction of the outcome for species 1 and 2. Fig. 8 Image Source adapted from Begon et al. (1996) & Gotelli (1998) 23 Fig. 9 Image Source adapted from Begon et al. (1996) & Gotelli (1998) Scenarios 3 and 4 can be considered to be made by alterations of the value for ​α​21, and ​α​12​.​ ​ In this case, both species’ isoclines intersect at a point. However in scenario 3, K​1​ is higher than K​2​ ​/ α​21​ ​, and K​2​ is higher than K​1​/ ​α​12​. ​The point of equilibrium is unstable, so the outcome of the species depends on the initial abundances of the two. The trajectories in between the isoclines diverge away from each other, and away from the point of equilibrium. In scenario 4, while the isoclines of both species intersect, their carrying capacities ​α​12, and ​α​21​ ​are lower ​ than K​2​ / ​α​21​ ​and K​1​/ ​α​12​.​ The population increases below and decreases above the ​ isoclines of both species. However in this case, the arrows in between the isoclines 24 of the two species tend to converge toward the intersection of the isoclines. Thus, the two species will coexist at the point of intersection, regardless of the conditions of the initial abundances. Henceforth, the idea behind the values of K​1​ and K​2​ is not well understood where both the sides of the line will intersect each other at a point. The arrows indicated on the graph show that they can move randomly in either direction. If the same technique was used to predict which species is the predator and which is the prey, the lengths of K​1​ and K​2​ are not fairly different. So in this kind of situation, as ​α​21​ ​and ​α​12​ ​are quite similar in lengths, it is more challenging and difficult to predict whether species 1 or species 2 will either win or lose. On that account where the two lines intersect at a point, it can be said that either species 1 or species 2 can win, as both are capable of happening. To conclude, that is how the outcome of two species can be predicted to either win or lose in certain situations, using the Lotka-Volterra model. 25 2.4 Conclusion To conclude, that is how the outcome of two species can be predicted to either win or lose in certain situations, using the Lotka-Volterra model. Several assumptions must be made in order to implement the constraints onto the equations for derivations. Those assumptions include factors such as the environment, changing conditions, competition of the species, as well as other ecological factors belonging to the surroundings. 26 References: - Boundless. (n.d.). Boundless Biology. 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